3.92 \(\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\)

Optimal. Leaf size=515 \[ \frac{a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac{a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35}{128} a^3 b^2 x-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac{9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{63 a^5 x}{256}-\frac{a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac{3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac{a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15}{256} a b^4 x+\frac{b^5 \sin ^{10}(c+d x)}{10 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^6(c+d x)}{6 d} \]

[Out]

(63*a^5*x)/256 + (35*a^3*b^2*x)/128 + (15*a*b^4*x)/256 - (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) - (a^4*b*Cos[c + d*x
]^10)/(2*d) + (a^2*b^3*Cos[c + d*x]^10)/d + (63*a^5*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (35*a^3*b^2*Cos[c + d
*x]*Sin[c + d*x])/(128*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (21*a^5*Cos[c + d*x]^3*Sin[c + d*x]
)/(128*d) + (35*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) +
(21*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) + (7*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^4*Cos[c
+ d*x]^5*Sin[c + d*x])/(32*d) + (9*a^5*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (a^3*b^2*Cos[c + d*x]^7*Sin[c + d
*x])/(8*d) - (3*a*b^4*Cos[c + d*x]^7*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^9*Sin[c + d*x])/(10*d) - (a^3*b^
2*Cos[c + d*x]^9*Sin[c + d*x])/d - (a*b^4*Cos[c + d*x]^7*Sin[c + d*x]^3)/(2*d) + (b^5*Sin[c + d*x]^6)/(6*d) -
(b^5*Sin[c + d*x]^8)/(4*d) + (b^5*Sin[c + d*x]^10)/(10*d)

________________________________________________________________________________________

Rubi [A]  time = 0.48451, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3090, 2635, 8, 2565, 30, 2568, 14, 2564, 266, 43} \[ \frac{a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac{a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35}{128} a^3 b^2 x-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac{9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{63 a^5 x}{256}-\frac{a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac{3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac{a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15}{256} a b^4 x+\frac{b^5 \sin ^{10}(c+d x)}{10 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(63*a^5*x)/256 + (35*a^3*b^2*x)/128 + (15*a*b^4*x)/256 - (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) - (a^4*b*Cos[c + d*x
]^10)/(2*d) + (a^2*b^3*Cos[c + d*x]^10)/d + (63*a^5*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (35*a^3*b^2*Cos[c + d
*x]*Sin[c + d*x])/(128*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (21*a^5*Cos[c + d*x]^3*Sin[c + d*x]
)/(128*d) + (35*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) +
(21*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) + (7*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^4*Cos[c
+ d*x]^5*Sin[c + d*x])/(32*d) + (9*a^5*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (a^3*b^2*Cos[c + d*x]^7*Sin[c + d
*x])/(8*d) - (3*a*b^4*Cos[c + d*x]^7*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^9*Sin[c + d*x])/(10*d) - (a^3*b^
2*Cos[c + d*x]^9*Sin[c + d*x])/d - (a*b^4*Cos[c + d*x]^7*Sin[c + d*x]^3)/(2*d) + (b^5*Sin[c + d*x]^6)/(6*d) -
(b^5*Sin[c + d*x]^8)/(4*d) + (b^5*Sin[c + d*x]^10)/(10*d)

Rule 3090

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Sym
bol] :> Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] &&
IntegerQ[m] && IGtQ[n, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos ^{10}(c+d x)+5 a^4 b \cos ^9(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^8(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^7(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^6(c+d x) \sin ^4(c+d x)+b^5 \cos ^5(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^5 \int \cos ^{10}(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^9(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^8(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^7(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^5(c+d x) \sin ^5(c+d x) \, dx\\ &=\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{1}{10} \left (9 a^5\right ) \int \cos ^8(c+d x) \, dx+\left (a^3 b^2\right ) \int \cos ^8(c+d x) \, dx+\frac{1}{2} \left (3 a b^4\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{\left (5 a^4 b\right ) \operatorname{Subst}\left (\int x^9 \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int x^7 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int x^5 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{1}{80} \left (63 a^5\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{8} \left (7 a^3 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{16} \left (3 a b^4\right ) \int \cos ^6(c+d x) \, dx-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (x^7-x^9\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int (1-x)^2 x^2 \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{1}{32} \left (21 a^5\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{48} \left (35 a^3 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{32} \left (5 a b^4\right ) \int \cos ^4(c+d x) \, dx+\frac{b^5 \operatorname{Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^{10}(c+d x)}{10 d}+\frac{1}{128} \left (63 a^5\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{64} \left (35 a^3 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{128} \left (15 a b^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^{10}(c+d x)}{10 d}+\frac{1}{256} \left (63 a^5\right ) \int 1 \, dx+\frac{1}{128} \left (35 a^3 b^2\right ) \int 1 \, dx+\frac{1}{256} \left (15 a b^4\right ) \int 1 \, dx\\ &=\frac{63 a^5 x}{256}+\frac{35}{128} a^3 b^2 x+\frac{15}{256} a b^4 x-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac{63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac{7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac{9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac{a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac{a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac{a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac{a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^{10}(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 1.26564, size = 307, normalized size = 0.6 \[ \frac{120 a \left (70 a^2 b^2+63 a^4+15 b^4\right ) (c+d x)+300 a \left (14 a^2 b^2+21 a^4+b^4\right ) \sin (2 (c+d x))+600 a \left (-2 a^2 b^2+3 a^4-b^4\right ) \sin (4 (c+d x))+50 a \left (-26 a^2 b^2+9 a^4-3 b^4\right ) \sin (6 (c+d x))+75 a \left (-6 a^2 b^2+a^4+b^4\right ) \sin (8 (c+d x))+6 a \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (10 (c+d x))-1200 a^2 b \left (3 a^2+b^2\right ) \cos (4 (c+d x))-300 a^2 b \left (a^2-b^2\right ) \cos (8 (c+d x))-300 b \left (14 a^2 b^2+21 a^4+b^4\right ) \cos (2 (c+d x))+50 b \left (6 a^2 b^2-27 a^4+b^4\right ) \cos (6 (c+d x))-6 b \left (-10 a^2 b^2+5 a^4+b^4\right ) \cos (10 (c+d x))}{30720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(120*a*(63*a^4 + 70*a^2*b^2 + 15*b^4)*(c + d*x) - 300*b*(21*a^4 + 14*a^2*b^2 + b^4)*Cos[2*(c + d*x)] - 1200*a^
2*b*(3*a^2 + b^2)*Cos[4*(c + d*x)] + 50*b*(-27*a^4 + 6*a^2*b^2 + b^4)*Cos[6*(c + d*x)] - 300*a^2*b*(a^2 - b^2)
*Cos[8*(c + d*x)] - 6*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[10*(c + d*x)] + 300*a*(21*a^4 + 14*a^2*b^2 + b^4)*Sin[2
*(c + d*x)] + 600*a*(3*a^4 - 2*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + 50*a*(9*a^4 - 26*a^2*b^2 - 3*b^4)*Sin[6*(c +
d*x)] + 75*a*(a^4 - 6*a^2*b^2 + b^4)*Sin[8*(c + d*x)] + 6*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[10*(c + d*x)])/(307
20*d)

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Maple [A]  time = 0.303, size = 335, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{10}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{60}} \right ) +5\,a{b}^{4} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +10\,{a}^{2}{b}^{3} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}-1/40\, \left ( \cos \left ( dx+c \right ) \right ) ^{8} \right ) +10\,{a}^{3}{b}^{2} \left ( -1/10\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{\sin \left ( dx+c \right ) }{80} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+7/6\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{7\,dx}{256}}+{\frac{7\,c}{256}} \right ) -{\frac{{a}^{4}b \left ( \cos \left ( dx+c \right ) \right ) ^{10}}{2}}+{a}^{5} \left ({\frac{\sin \left ( dx+c \right ) }{10} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{21\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{64}}+{\frac{315\,\cos \left ( dx+c \right ) }{128}} \right ) }+{\frac{63\,dx}{256}}+{\frac{63\,c}{256}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)

[Out]

1/d*(b^5*(-1/10*sin(d*x+c)^4*cos(d*x+c)^6-1/20*sin(d*x+c)^2*cos(d*x+c)^6-1/60*cos(d*x+c)^6)+5*a*b^4*(-1/10*sin
(d*x+c)^3*cos(d*x+c)^7-3/80*sin(d*x+c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(
d*x+c)+3/256*d*x+3/256*c)+10*a^2*b^3*(-1/10*sin(d*x+c)^2*cos(d*x+c)^8-1/40*cos(d*x+c)^8)+10*a^3*b^2*(-1/10*sin
(d*x+c)*cos(d*x+c)^9+1/80*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+7/256
*d*x+7/256*c)-1/2*a^4*b*cos(d*x+c)^10+a^5*(1/10*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16*cos(d*x+c)^5+105/64*cos(d
*x+c)^3+315/128*cos(d*x+c))*sin(d*x+c)+63/256*d*x+63/256*c))

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Maxima [A]  time = 1.23067, size = 392, normalized size = 0.76 \begin{align*} -\frac{15360 \, a^{4} b \cos \left (d x + c\right )^{10} - 3 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 2520 \, d x + 2520 \, c + 25 \, \sin \left (8 \, d x + 8 \, c\right ) + 600 \, \sin \left (4 \, d x + 4 \, c\right ) + 2560 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} + 10 \,{\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c + 45 \, \sin \left (8 \, d x + 8 \, c\right ) + 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} + 7680 \,{\left (4 \, \sin \left (d x + c\right )^{10} - 15 \, \sin \left (d x + c\right )^{8} + 20 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} - 512 \,{\left (6 \, \sin \left (d x + c\right )^{10} - 15 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{30720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/30720*(15360*a^4*b*cos(d*x + c)^10 - 3*(32*sin(2*d*x + 2*c)^5 - 640*sin(2*d*x + 2*c)^3 + 2520*d*x + 2520*c
+ 25*sin(8*d*x + 8*c) + 600*sin(4*d*x + 4*c) + 2560*sin(2*d*x + 2*c))*a^5 + 10*(96*sin(2*d*x + 2*c)^5 - 640*si
n(2*d*x + 2*c)^3 - 840*d*x - 840*c + 45*sin(8*d*x + 8*c) + 120*sin(4*d*x + 4*c))*a^3*b^2 + 7680*(4*sin(d*x + c
)^10 - 15*sin(d*x + c)^8 + 20*sin(d*x + c)^6 - 10*sin(d*x + c)^4)*a^2*b^3 - 15*(32*sin(2*d*x + 2*c)^5 + 120*d*
x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a*b^4 - 512*(6*sin(d*x + c)^10 - 15*sin(d*x + c)^8 + 10*
sin(d*x + c)^6)*b^5)/d

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Fricas [A]  time = 0.605827, size = 595, normalized size = 1.16 \begin{align*} -\frac{640 \, b^{5} \cos \left (d x + c\right )^{6} + 384 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{10} + 960 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{8} - 15 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x -{\left (384 \,{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 48 \,{\left (9 \, a^{5} + 10 \, a^{3} b^{2} - 55 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

-1/3840*(640*b^5*cos(d*x + c)^6 + 384*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^10 + 960*(5*a^2*b^3 - b^5)*cos
(d*x + c)^8 - 15*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*d*x - (384*(a^5 - 10*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^9 + 48*
(9*a^5 + 10*a^3*b^2 - 55*a*b^4)*cos(d*x + c)^7 + 8*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 + 10*(63*a^
5 + 70*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^3 + 15*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 48.4455, size = 979, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Piecewise((63*a**5*x*sin(c + d*x)**10/256 + 315*a**5*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 315*a**5*x*sin(c
+ d*x)**6*cos(c + d*x)**4/128 + 315*a**5*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 315*a**5*x*sin(c + d*x)**2*co
s(c + d*x)**8/256 + 63*a**5*x*cos(c + d*x)**10/256 + 63*a**5*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 147*a**5*s
in(c + d*x)**7*cos(c + d*x)**3/(128*d) + 21*a**5*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 237*a**5*sin(c + d*x
)**3*cos(c + d*x)**7/(128*d) + 193*a**5*sin(c + d*x)*cos(c + d*x)**9/(256*d) - a**4*b*cos(c + d*x)**10/(2*d) +
35*a**3*b**2*x*sin(c + d*x)**10/128 + 175*a**3*b**2*x*sin(c + d*x)**8*cos(c + d*x)**2/128 + 175*a**3*b**2*x*s
in(c + d*x)**6*cos(c + d*x)**4/64 + 175*a**3*b**2*x*sin(c + d*x)**4*cos(c + d*x)**6/64 + 175*a**3*b**2*x*sin(c
+ d*x)**2*cos(c + d*x)**8/128 + 35*a**3*b**2*x*cos(c + d*x)**10/128 + 35*a**3*b**2*sin(c + d*x)**9*cos(c + d*
x)/(128*d) + 245*a**3*b**2*sin(c + d*x)**7*cos(c + d*x)**3/(192*d) + 7*a**3*b**2*sin(c + d*x)**5*cos(c + d*x)*
*5/(3*d) + 395*a**3*b**2*sin(c + d*x)**3*cos(c + d*x)**7/(192*d) - 35*a**3*b**2*sin(c + d*x)*cos(c + d*x)**9/(
128*d) - 5*a**2*b**3*sin(c + d*x)**2*cos(c + d*x)**8/(4*d) - a**2*b**3*cos(c + d*x)**10/(4*d) + 15*a*b**4*x*si
n(c + d*x)**10/256 + 75*a*b**4*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 75*a*b**4*x*sin(c + d*x)**6*cos(c + d*x
)**4/128 + 75*a*b**4*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 75*a*b**4*x*sin(c + d*x)**2*cos(c + d*x)**8/256 +
15*a*b**4*x*cos(c + d*x)**10/256 + 15*a*b**4*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 35*a*b**4*sin(c + d*x)**7
*cos(c + d*x)**3/(128*d) + a*b**4*sin(c + d*x)**5*cos(c + d*x)**5/(2*d) - 35*a*b**4*sin(c + d*x)**3*cos(c + d*
x)**7/(128*d) - 15*a*b**4*sin(c + d*x)*cos(c + d*x)**9/(256*d) - b**5*sin(c + d*x)**4*cos(c + d*x)**6/(6*d) -
b**5*sin(c + d*x)**2*cos(c + d*x)**8/(12*d) - b**5*cos(c + d*x)**10/(60*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c)
)**5*cos(c)**5, True))

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Giac [A]  time = 1.37491, size = 462, normalized size = 0.9 \begin{align*} \frac{1}{256} \,{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x - \frac{{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac{5 \,{\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac{5 \,{\left (27 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{5 \,{\left (3 \, a^{4} b + a^{2} b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac{5 \,{\left (21 \, a^{4} b + 14 \, a^{2} b^{3} + b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac{{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{5 \,{\left (a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac{5 \,{\left (9 \, a^{5} - 26 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} + \frac{5 \,{\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{5 \,{\left (21 \, a^{5} + 14 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/256*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*x - 1/5120*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(10*d*x + 10*c)/d - 5/512*(a
^4*b - a^2*b^3)*cos(8*d*x + 8*c)/d - 5/3072*(27*a^4*b - 6*a^2*b^3 - b^5)*cos(6*d*x + 6*c)/d - 5/128*(3*a^4*b +
a^2*b^3)*cos(4*d*x + 4*c)/d - 5/512*(21*a^4*b + 14*a^2*b^3 + b^5)*cos(2*d*x + 2*c)/d + 1/5120*(a^5 - 10*a^3*b
^2 + 5*a*b^4)*sin(10*d*x + 10*c)/d + 5/2048*(a^5 - 6*a^3*b^2 + a*b^4)*sin(8*d*x + 8*c)/d + 5/3072*(9*a^5 - 26*
a^3*b^2 - 3*a*b^4)*sin(6*d*x + 6*c)/d + 5/256*(3*a^5 - 2*a^3*b^2 - a*b^4)*sin(4*d*x + 4*c)/d + 5/512*(21*a^5 +
14*a^3*b^2 + a*b^4)*sin(2*d*x + 2*c)/d